A linear mapping from a vector space to a field of scalars. In other words, a linear function which acts upon a vector resulting in a real number (scalar)
\begin{equation} \alpha\,:\,\mathbf{V} \longrightarrow \mathbb{R} \end{equation}
Simplistically, covectors can be thought of as “row vectors”, or:
\begin{equation} \begin{bmatrix} 1 & 2 \end{bmatrix} \end{equation}
This might look like a standard vector, which would be true in an orthonormal basis, but it is not true generally.
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Tensors Tensor Product

A mapping from \(\mathbf{V} \rightarrow \mathbf{W}\) that preserves the operations of addition and scalar multiplication.
Also known as # Linear Map Linear Transformation Linear Function

A function of several variables that is linear, separately, in each variable.
A multilinear map of one variable is a standard linear mapping.

The space of all linear functionals \(f:V\rightarrow \mathbb{R}\), noted as \(V^{*}\)
The dual space has the same dimension as the corresponding vector space or, given a space \(V\), with bases \((v_{1},…,v_{n})\), there exists a dual space \(V^{*}\) with a dual basis \((v^{*}_{1},…,v^{*}_{n})\).

The space of all linear functionals \(f:V\rightarrow \mathbb{R}\), noted as \(V^{*}\)
The dual space has the same dimension as the corresponding vector space or, given a space \(V\), with bases \((v_{1},…,v_{n})\), there exists a dual space \(V^{*}\) with a dual basis \((v^{*}_{1},…,v^{*}_{n})\).

As a linear representation # A tensor can be represented as a vector of x-number of dimensions. Basically, a generalization on top of scalars, vectors, and matrices. The specific “flavor” of the tensor (i.e. is it a scalar, vector, or matrix) is clarified by referring to the tensor’s “rank”. For instance; a rank 0 tensor is a scalr, rank 1 tensor is a one-dimensional vector, a rank 2 tensor is a two-dimensional vector (\(2x2\) matrix), etc.
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a basis for an n-dimensional vector space \(V\) is any ordered set of linearly independent vectors \((\mathbf{e}_{1}, \mathbf{e}_{2},…,\mathbf{e}_{n})\)
An arbitrary vector \(\mathbf{x}\) in \(V\) can be expressed as a linear combination of the basis vectors:
\begin{equation} \mathbf{x}\,=\,\sum\limits_{i = 1}^{n} \mathbf{e}_{i}x^{i} \end{equation}
See Bases Transformation, Coordinate Transformation

An orthonormal basis is a basis where all the vectors are one unit long and all perpendicular to each other (e.g. the Cartesian plane)